Stationary solutions of retarded Ornstein–Uhlenbeck processes in Hilbert spaces
Introduction
Let and be two complex separable Hilbert spaces and a probability space equipped with some right-continuous filtration such that contains all sets of -measure zero. Suppose that is a -valued, -Wiener process on with respect to (see Da Prato and Zabczyk (1992)) with covariance operator which is a linear, symmetric and nonnegative bounded operator on . If the trace Tr , then is a genuine Wiener process. It is possible that Tr , e.g., which corresponds to a cylindrical process. For arbitrary , consider the so-called Langevin’s equation on the Hilbert space where is the infinitesimal generator of a -semigroup , , and , the space of all bounded and linear operators from into . One can show that there exists a unique mild solution to (1.1) if the operators are of trace class for any (see Da Prato and Zabczyk (1996)). Moreover, in this case the solution is given by the formula The process is called an Ornstein–Uhlenbeck (O–U) process which is Gaussian and Markovian. The Markovian property of makes it useful to introduce the Markovian transition function , , , the Borel -field on , and associated Markovian transition semigroup , , on the family of all bounded, continuous functions on the Hilbert space by and further the invariant (probability) measure with respect to if and only if
There exists extensive literature on invariant measures, stationary solutions and related topics about the system (1.1), e.g., see Chojnowska-Michalik (1987) and Da Prato and Zabczyk, 1992, Da Prato and Zabczyk, 1996 and reference cited therein. For instance, it is shown that if is an invariant measure and, in addition, the initial distribution of the solution of (1.1) with random initial is , then the process is stationary.
In this note, we are concerned about the stationary (mild) solutions of the following stochastic retarded differential equation on the Hilbert space , for properly given random initial . Here , and , and is a linear, generally unbounded, operator having the property that is bounded so that there exists an -valued Stieltjes measure such that When one tries to investigate the large time behaviour of the stochastic retarded system (1.4), one of the main difficulties is that the solution , unlike the system (1.1), would not be Markovian any more. As an immediate consequence, the important concepts of Markovian transition functions, semigroups, etc. analogous to (1.2), (1.3) would make less sense in the study of the stochastic retarded system (1.4). In this work, we shall present a decent description of the stationary solutions of (1.4) by means of Green operators, Lyapunov equations, the second moment bounds or even retarded point spectrum when generates a compact semigroup on . It should be mentioned that stationary solutions were studied in Küchler and Mensch (1992) for a class of one-dimensional Langevin stochastic differential equations with time delays.
Before moving to a detailed investigation of the stochastic retarded system (1.4), let us first review some material from deterministic theory. For any fixed constant , we denote by the Hilbert space of all -valued equivalence classes of measurable functions which are square integrable on . Let denote the product Hilbert space with the norm Consider the following retarded deterministic evolution equation on , where is given as in (1.4), and its corresponding integral equation It has been shown (see Liu (in press)) that for any , the Eq. (1.7) has a unique solution which is called the mild solution of (1.6). In particular, for any , let , for and . One can define the fundamental solutions or Green operators of (1.6) with this initial datum by The definition (1.8) implies that satisfies the integral equation where , , and denotes the null operator on . It may be shown that is a strongly continuous one-parameter family of bounded linear operators on such that , , for some constants and . Moreover, the relation holds almost surely (see Liu (in press)).
In some important topics, for instance, optimal control theory, it is quite useful to establish an adjoint theory for the system (1.6). To this end, we first remark that in association with the above operator , for any , one may define a linear and bounded operator by for any . Unless otherwise specified, we always identity with its adjoint space in this work. Let denote the adjoint operator of . It is possible to show that there exists a unique operator to such that is linear and bounded from to and for any , where represents the complex conjugate of .
Let and define the “formal” transposed retarded system of (1.6) on by where denote the adjoint operator of . It is well known that generates a -semigroup on , which is the adjoint of , . Hence, we can construct a fundamental solution which is characterized as the unique solution of We denote by the adjoint of , , and it may be shown (see Liu (in press) or Nakagiri (1986)) that for all . Moreover, it is valid that if , then for each , and is absolutely continuous such that where is the adjoint operator of , .
The mild solution of (1.6) is closely related to a -semigroup on the product Hilbert space . Actually, let us define a mapping , , associated with of (1.6), by where for any and . It can be shown that the mapping , , is a -semigroup with some infinitesimal generator on the space . Particularly, the operator is explicitly characterized by the following proposition. Let denote the Sobolev space of all -valued on such that and its distributional derivative belong to . Proposition 1.1 The generator of the -semigroup , denote it by , is described by
For each , we define the densely defined closed linear operator by The retarded resolvent set is defined as the set of all values in for which the operator has a bounded inverse, denoted by , on . The retarded spectrum is defined to be . In particular, , the retarded point spectrum of , if and only if there exists a nonzero such that . The value is often called the characteristic value of .
Section snippets
Stationary solutions
Let denote the space of all -valued mappings defined on such that both and are -measurable for any and satisfy We shall be concerned about the following stochastic retarded evolution equation on the Hilbert space , where , is a -valued -Wiener process on and is given as in (1.4).
Definition 2.1 A stochastic processes
Acknowledgement
The author wishes to express his sincere thanks for the referee’s constructive comments on this work.
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2009, Statistics and Probability LettersCitation Excerpt :On the other hand, there are a few papers already published for the stationary solution of stochastic partial differential equations. For example, Bakhtin and Mattingly (2005) proved the existence and uniqueness of the stationary solution of stochastically forced dissipative partial differential equations such as the stochastic Navier–Stokes equation and stochastic Ginsburg–Landau equation; Bessaih (2008) studied the existence of stationary martingale solutions of a two-dimensional dissipative Euler equation subjected to a random perturbation; Blömker and Hairer (2004) considered the existence of stationary mild solutions for a model of amorphous thin-film growth; Caraballo et al. (2004) considered the existence of a non-trivial exponentially stable stationary solution of stochastic evolution equations on some separable Hilbert space; Liu (2008a,b) considered the stationary solution of retarded Ornstein–Uhlenbeck processes in Hilbert spaces; Liu (2009) studied the stationary solution of a class of Langevin equations driven by Lévy processes with time delays. Our second aim in the present paper is to deal with the stationary mild solutions of Eq. (1.1) with multiplicative noise.
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